1,1

These pentagonal numbers P(k) that can be represented as the sum of P(i)+j^2, i,j>0, are at k= 2, 6, 9, 10, 13, 17, 21, 22, 24, 26, 29, 34, 35, 38, 41, 45, 46, 53. Are almost all positive integers in this sequence and, if so, what is the largest value in the complement? The largest square in the complement? The largest pentagonal number in the complement?

{A000326(i) + A000290(j) for i, j > 0}. {i(3*i-1)/2 + j^2 for i, j > 0}.

Let P(n) = n-th pentagonal number:

a(1) = P(1) + 1^2 = 1 + 1 = 2.

a(2) = P(1) + 2^2 = 1 + 4 = 5 = P(2).

a(3) = P(2) + 1^2 = 5 + 1 = 6.

a(4) = P(2) + 2^2 = 5 + 4 = 9 = 3^2.

a(5) = P(1) + 3^2 = 1 + 9 = 10 = a(P(2)).

a(8) = P(3) + 2^2 = 12 + 4 = 16 = 4^2.

a(10) = P(2) + 4^2 = 5 + 16 = P(3) + 3^2 = 12 + 9 = 21.

a(12) = P(1) + 5^2 = 1 + 25 = P(4) + 2^2 = 22 + 4 = 26 = a(P(3)).

a(16) = P(5) + 1^2 = 35 + 1 = 36 = 6^2.

a(17) = P(1) + 6^2 = 1 + 36 = P(3) + 5^2 = 12 + 25 = 37.

a(25) = P(5) + 4^2 = 35 + 16 = 51 = P(6).

a(30) = P(6) + 3^2 = 51 + 9 = P(5) + 5^2 = 35 + 25 = 60.

a(35) = P(7) + 1^2 = 70 + 1 = P(5) + 6^2 = 35 + 36 = P(4) + 7^2 = 22 + 49 = 71 = a(P(5)).

a(37) = P(6) + 5^2 = 51 + 25 = P(3) + 8^2 = 12 + 64 = 76.

a(41) = P(7) + 4^2 = 70 + 16 = P(4) + 8^2 = 22 + 64 = 86.

Cf. A000290, A000326, A134935-A134938.

Sequence in context: A028946 A031461 A085183 this_sequence A042963 A166097 A000277

Adjacent sequences: A133756 A133757 A133758 this_sequence A133760 A133761 A133762

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 21 2008

Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 21 2008